Signs and symbolism
Of all the subjects in the curriculum, mathematics probably requires the greatest precision in the use of language and yet it is probably the least well served in this respect by teachers at all levels.
The problems arise from the two ends of the comprehension spectrum. At one extreme there are those who have always been good at maths and can't understand that there can be any difficulties in the subject. At the other extreme there are those who have "always had a fear of maths" and tend to teach as they were taught however inadequate that teaching may have been. Frequently the best teachers are those who have themselves struggled with the subject, recognising and eventually satisfactorily resolving the problems.
There are several kinds of error: 1: The use of slang or "baby talk" When children arrive in Reception class able to get their tongues round diplodocus and ichthyosaur, teachers who would not dream of talking about bow-wows and chuff-chuffs need not stoop to "four times two". There is no such verb as "to times". "Multiply" and "multiplication" should hold no terrors.
2: Lack of understanding on the part of the teacher For example, some primary teachers call division sums "sharing" when many situations are not division but "grouping". Compare the expected response 12 V 3 = 4 with (a) 12 sweets are divided between three children. How many does each receive?
(b) Mary has 12 sweets. She gives 3 to each of her friends and has none left. How many friends has she?
Some other situations are neither sharing nor grouping (for example, comparison of lengths). It is this multiplicity of applications to the sign "V" which creates difficulty. Similarly, most teachers over 50 (and a good few younger) were taught to do subtraction sums by "borrowing" and "paying back". Analysis of the method in question should send a shiver down any teacher's spine, yet it persists in places.
3: Where mathematical language conflicts with its everyday use Reading 8 V 2 = 4 as "2 into 8 goes 4", or even "if you divide 2 into 8 it goes 4 times" is confusing since, as every child knows, if you divide 2 bars of chocolate into 8 you don't get 4 bars of chocolate. You get a quarter of a bar of chocolate each.
Is it any wonder then that many children feel that mathematics is divorced from the real world, or that right up to GCSE the sign "V" is commonly interpreted as "into"; naturally, statements such as 4 V 32 = 8 are common.
4: Different language used in the same situation 3 x 2 = 6 can be read as: (a) 3 multiplied by 2 = 6 (b) 3 times 2 = 6 (c) 2 threes are 6, or (d) 3 twos are 6 To a child these are all different even if (a)(c) and (b)(d) could be considered equivalent pairs. And it is no defence to say that it doesn't matter since 2 threes and 3 twos are the same. They are not.
To a child, taking two objects and tripling is not the same thing as taking three objects and doubling. The subtlety of commutativity of addition and multiplication and non-commutativity of subtraction and division is a very real one. Indeed, the non-commutativity of division is made almost incomprehensible by the example in 3 above.
5: Everyday mathematical language in conflict with mathematical desirability Although there is a variety of usage among teachers of maths, it is confusing to children. For example, numbers are divided into positive, negative and zero, but unfortunately the "negative" have traditionally been called "minus" - for example, we speak of temperatures of minus 5. So it is quite likely for "7-(-2)" to be read as "7 minus minus 2" rather than "7 minus negative 2". The confusion arises because in the first reading the "minus" has two different meanings - as an operator and as a label. It is preferable for teachers to refer to "negative 2" consistently and to point out to pupils that they may well meet "minus 2" in other situations.
Another problem arises with the choice between "square centimetre" and "centimetre squared". "Centimetre squared" at least suggests the process by which the area of a square is found.
6: Many words have more than one meaning If they didn't, compilers of crossword puzzles would be out of business. There are two categories of which teachers need to be aware: (a) some words have a special meaning in mathematics different from their meaning in everyday usage, for example face, table, flat, left, top, row, index.
(b) some words have more than one meaning within mathematics, for example base, square, rule, score.
It is essential, therefore, that new definitions are carefully introduced and the distinction drawn between these and existing definitions.
For children a further complication is the inconsistency between teachers as a child moves up the school. One subset of the mathematical language is suddenly replaced by another without explanation and the child has to effect the translation unaided.
If these problems are accepted it would seem beneficial for both teachers and pupils if a common language were to be developed. In some instances the choice is clear. In other cases it is a matter of judgment. Once a standard language had been agreed, it would have to be adhered to as far as humanly possible.
Consistency would transform primary maths teaching. Let's look at the "four rules". Consider: 6 + 2 = 8 6 - 2 = 4 6 x 2 = 12 6 V 2 = 3 If in each case the 6 is taken as the starting point and is followed, in order, by (i) the operation to be performed and (ii) the "extent" of that operation, then suggested readings of these statements would be: six plus two equals eight six minus two equals four six multiplied by two equals twelve six divided by two equals three Give the operation signs correct names from the outset. Perhaps there might just be a case for "six add two" and "six take away two" in the early stages, but the correct terminology can be quickly substituted once the process is understood.
Multiplication, if applied consistently from the outset, will present no difficulty, but division in its various manifestations will always require careful handling.
There are implications for "tables", since the only helpful tables are tables of twos, threes, fours, etc. For example: one three is three 3 x 1 = 3 two threes are six 3 x 2 = 6 three threes are nine 3 x 3 = 9 four three are twelve 3 x 4 = 12 comes naturally from collecting sets of three. The "times" table, however, is redundant. Look at: three times one is three three times two is six three times three is nine three times four is twelve.
One crucial area is the teaching of fractions. Few children would have difficulty in appreciating that one dog plus one dog equals two dogs, but the same children meeting l4 + l4 = 24 let alone l4 + l4 = l2 are likely to be much less secure.
Recording experience as one quarter plus one quarter equals two quarters for some time before introducing the notation is likely to be very beneficial.
Sometimes as teachers we forget how much we expect of children in the way of symbolism and how powerful our succinct use of symbols is. Consider for example the digits 2 and 3. If the 2 gyrates round the 3 the pair has a different meaning every 45x: 23, 32, 32, 32, 32, 23, 23, 23 even allowing for the questionable use of 32 These suggestions may appear daunting and it is true that at first the discipline required to sustain a consistent approach may be difficult, but the benefits for both teachers and children will be immeasurable.
Stuart Parker is former senior lecturer in mathematics at Hertfordshire College of Higher Education, now University of Hertfordshire. He was also chief examiner for GCE with the AEB and still examines for its successor, the SEG